Find Vector Projection Calculator

Calculate the projection of vector A onto vector B.

Vector A (the vector being projected):

Vector B (the vector being projected onto):

Results copied!

What is a Vector Projection?

A vector projection (or vector resolute) of a vector a onto a non-zero vector b is the orthogonal projection of a onto a straight line parallel to b. Essentially, it’s like finding the “shadow” that vector a casts onto the line defined by vector b if a light source were perpendicular to b. The result is a vector that lies along the direction of b (or opposite to it), representing the component of a in the direction of b. Our find vector projection calculator helps you compute this easily.

The projection of vector a onto b is denoted as proj_b a. It is itself a vector.

Who should use it? Students of physics, engineering, mathematics, and computer graphics often need to calculate vector projections. It’s used in mechanics to find components of forces along certain directions, in computer graphics for lighting and shadow calculations, and in various areas of linear algebra and geometry. The find vector projection calculator is a handy tool for these applications.

Common Misconceptions:

• Scalar vs. Vector Projection: The scalar projection (or component) is just the length of the vector projection (with a sign indicating direction), while the vector projection is the vector itself. Our find vector projection calculator gives you both.
• Direction: The vector projection always lies along the line defined by vector b, not a.

Vector Projection Formula and Mathematical Explanation

The formula to find the vector projection of vector a onto vector b is:

proj_b a = ((a · b) / |b|²) * b

Where:

• a · b is the dot product of vectors a and b. If a = <a_x, a_y, a_z> and b = <b_x, b_y, b_z>, then a · b = a_xb_x + a_yb_y + a_zb_z.
• |b|² is the square of the magnitude (length) of vector b. |b|² = b_x² + b_y² + b_z². The magnitude |b| = √(b_x² + b_y² + b_z²).
• The term (a · b) / |b| is the scalar projection of a onto b, which is the signed magnitude of the vector projection.
• Multiplying this scalar by the unit vector in the direction of b (which is b / |b|) gives the vector projection: ((a · b) / |b|) * (b / |b|) = ((a · b) / |b|²) * b.

The find vector projection calculator implements this formula.

Variables Table

Variable	Meaning	Unit	Typical Range
a = <ax, ay, az>	The vector being projected	Depends on context (e.g., meters, N)	Real numbers
b = <bx, by, bz>	The vector onto which a is projected	Depends on context (e.g., meters, N)	Real numbers (non-zero for projection)
a · b	Dot product of a and b	Scalar (units of a * units of b)	Real numbers
|b|²	Magnitude of b squared	(Units of b)²	Positive real numbers (or zero if b is the zero vector)
projb a	Vector projection of a onto b	Same units as a and b	Vector with real number components

Practical Examples (Real-World Use Cases)

Example 1: Force Component

Imagine a force vector F = <3, 4, 0> N acting on an object, and we want to find the component of this force along a direction given by vector d = <1, 1, 0> m.

Using the find vector projection calculator or manually:

• a = F = <3, 4, 0>
• b = d = <1, 1, 0>
• F · d = (3)(1) + (4)(1) + (0)(0) = 3 + 4 = 7
• |d|² = 1² + 1² + 0² = 1 + 1 = 2
• proj_d F = (7 / 2) * <1, 1, 0> = <3.5, 3.5, 0> N

So, the component of the force F along the direction of d is <3.5, 3.5, 0> N.

Example 2: Work Done

If a constant force F = <10, -5, 2> N moves an object along a displacement vector s = <2, 3, 1> m, the work done is the dot product F · s. However, if we wanted to find the part of the force vector that acts along the displacement, we’d find the projection of F onto s.

• a = F = <10, -5, 2>
• b = s = <2, 3, 1>
• F · s = (10)(2) + (-5)(3) + (2)(1) = 20 – 15 + 2 = 7
• |s|² = 2² + 3² + 1² = 4 + 9 + 1 = 14
• proj_s F = (7 / 14) * <2, 3, 1> = 0.5 * <2, 3, 1> = <1, 1.5, 0.5> N

The vector <1, 1.5, 0.5> N is the component of the force F that acts in the direction of the displacement s. The find vector projection calculator quickly gives this result.

How to Use This Find Vector Projection Calculator

Our find vector projection calculator is straightforward to use:

1. Input Vector A: Enter the x (i), y (j), and z (k) components of the vector you want to project (Vector A).
2. Input Vector B: Enter the x (i), y (j), and z (k) components of the vector you are projecting onto (Vector B). Ensure Vector B is not the zero vector (<0, 0, 0>), as projection onto a zero vector is undefined.
3. View Results: The calculator automatically updates and displays:
   • The primary result: The projection vector (proj_B A) components.
   • Intermediate values: The dot product (A · B), the magnitude of B squared (|B|²), and the scalar projection.
4. Visualization: The chart shows a 2D (x-y plane) representation of vector A, vector B, and the projection vector for easier understanding.
5. Reset: Click “Reset” to clear the fields to default values.
6. Copy: Click “Copy Results” to copy the main results and inputs to your clipboard.

The find vector projection calculator provides immediate feedback as you change the input values.

Key Factors That Affect Vector Projection Results

Several factors influence the outcome of a vector projection:

1. Components of Vector A: The magnitude and direction of the vector being projected directly determine what is being projected.
2. Components of Vector B: The direction of vector B defines the line onto which vector A is projected. Its magnitude also affects the projection vector’s length through the |B|² term.
3. Angle Between Vectors A and B: The dot product A · B = |A||B|cos(θ), where θ is the angle between A and B. If θ = 90° (vectors are orthogonal), the dot product is 0, and the projection is the zero vector. If θ = 0° or 180°, the projection is maximal in magnitude.
4. Magnitude of Vector B: While the direction of the projection is along B, its magnitude is scaled by (A · B) / |B|². If |B| is very large, the scalar part can be small, but the vector B itself is large, so the components of the projection depend on this interplay. However, the scalar projection (A · B) / |B| is inversely proportional to |B|.
5. Zero Vector B: If vector B is the zero vector (<0, 0, 0>), its magnitude is zero, and projection is undefined because you cannot divide by zero. Our find vector projection calculator handles this.
6. Dimensionality: Whether you are working in 2D or 3D affects the number of components for each vector, but the formula remains the same. The calculator here is for 3D vectors (you can use it for 2D by setting z-components to 0).

Frequently Asked Questions (FAQ)

What is the difference between scalar and vector projection?

The scalar projection is a number representing the signed length of the shadow of vector A on vector B. The vector projection is a vector that has this length and points along the direction of vector B (or opposite if the scalar projection is negative). The find vector projection calculator shows both.

What happens if vector B is the zero vector?

Projection onto the zero vector is undefined because it involves division by the magnitude of B squared, which would be zero. The calculator will indicate an issue or return zero/NaN if B is zero.

Can the projection vector be longer than vector A?

No, the magnitude of the projection of A onto B is |A| |cos(θ)|, where θ is the angle between A and B. Since |cos(θ)| ≤ 1, the magnitude of the projection is always less than or equal to the magnitude of A.

What if vectors A and B are orthogonal?

If A and B are orthogonal (perpendicular), their dot product is zero, and the projection of A onto B is the zero vector (<0, 0, 0>). The find vector projection calculator will show this.

What if vectors A and B are parallel?

If A and B are parallel, the projection of A onto B is A itself (if they point in the same direction) or -A (if they point in opposite directions and B is used as the direction reference, though it’s more accurate to say the projection’s magnitude is |A| and direction along B).

Does the order matter (projection of A onto B vs. B onto A)?

Yes, proj_B A is generally different from proj_A B. They lie along different directions (B and A, respectively) and usually have different magnitudes.

How is the find vector projection calculator useful in physics?

It’s used to find components of forces, velocities, or accelerations along specific directions, or to calculate work done when force and displacement are vectors.

Can I use this find vector projection calculator for 2D vectors?

Yes, simply set the z-components (A_z and B_z) of both vectors to zero.

Related Tools and Internal Resources

• Vector Calculator: Perform various vector operations like addition, subtraction, and finding magnitude.
• Dot Product Calculator: Calculate the dot product of two vectors, a key part of finding the vector projection.
• Vector Magnitude Calculator: Find the length (magnitude) of a vector.
• Cross Product Calculator: Calculate the cross product of two vectors.
• Scalar Projection Explained: Learn more about the scalar component of the projection.
• Vector Components Guide: Understand the i, j, k components of vectors.